1 Set Operations CS/APMA 202, Spring 2005 Rosen, section 1.7 Aaron Bloomfield

2 Triangle shows mixable color range (gamut) – the set of colors Sets of Colors Monitor gamut (M) Printer gamut (P) Pick any 3 “primary” colors

3 A union of the sets contains all the elements in EITHER set Union symbol is usually a U Example: C = M U P Monitor gamut (M) Printer gamut (P) Set operations: Union 1

4 Set operations: Union 2 U AB A U B

5 Set operations: Union 3 Formal definition for the union of two sets: A U B = { x | x  A or x  B } Further examples –{1, 2, 3} U {3, 4, 5} = {1, 2, 3, 4, 5} –{New York, Washington} U {3, 4} = {New York, Washington, 3, 4} –{1, 2} U  = {1, 2}

6 Set operations: Union 4 Properties of the union operation –A U  = AIdentity law –A U U = U Domination law –A U A = AIdempotent law –A U B = B U ACommutative law –A U (B U C) = (A U B) U CAssociative law

7 An intersection of the sets contains all the elements in BOTH sets Intersection symbol is a ∩ Example: C = M ∩ P Monitor gamut (M) Printer gamut (P) Set operations: Intersection 1

8 Set operations: Intersection 2 U BA A ∩ B

9 Set operations: Intersection 3 Formal definition for the intersection of two sets: A ∩ B = { x | x  A and x  B } Further examples –{1, 2, 3} ∩ {3, 4, 5} = {3} –{New York, Washington} ∩ {3, 4} =  No elements in common –{1, 2} ∩  =  Any set intersection with the empty set yields the empty set

10 End of lecture on 1 February 2005

11 Set operations: Intersection 4 Properties of the intersection operation –A ∩ U = AIdentity law –A ∩  =  Domination law –A ∩ A = AIdempotent law –A ∩ B = B ∩ ACommutative law –A ∩ (B ∩ C) = (A ∩ B) ∩ CAssociative law

12 Disjoint sets 1 Two sets are disjoint if the have NO elements in common Formally, two sets are disjoint if their intersection is the empty set Another example: the set of the even numbers and the set of the odd numbers

13 Disjoint sets 2 U AB

14 Disjoint sets 3 Formal definition for disjoint sets: two sets are disjoint if their intersection is the empty set Further examples –{1, 2, 3} and {3, 4, 5} are not disjoint –{New York, Washington} and {3, 4} are disjoint –{1, 2} and  are disjoint Their intersection is the empty set –  and  are disjoint! Their intersection is the empty set

15 Set operations: Difference 1 A difference of two sets is the elements in one set that are NOT in the other Difference symbol is a minus sign Example: C = M - P Monitor gamut (M) Printer gamut (P) Also visa-versa: C = P - M

16 Set operations: Difference 2 U AB B - A A - B

17 Formal definition for the difference of two sets: A - B = { x | x  A and x  B } A - B = A ∩ B  Important! Further examples –{1, 2, 3} - {3, 4, 5} = {1, 2} –{New York, Washington} - {3, 4} = {New York, Washington} –{1, 2} -  = {1, 2} The difference of any set S with the empty set will be the set S Set operations: Difference 3 _

18 A symmetric difference of the sets contains all the elements in either set but NOT both Symetric diff. symbol is a  Example: C = M  P Monitor gamut (M) Printer gamut (P) Set operations: Symmetric Difference 1

19 Formal definition for the symmetric difference of two sets: A  B = { x | (x  A or x  B) and x  A ∩ B} A  B = (A U B) – (A ∩ B)  Important! Further examples –{1, 2, 3}  {3, 4, 5} = {1, 2, 4, 5} –{New York, Washington}  {3, 4} = {New York, Washington, 3, 4} –{1, 2}   = {1, 2} The symmetric difference of any set S with the empty set will be the set S Set operations: Symmetric Difference 2

20 A complement of a set is all the elements that are NOT in the set Difference symbol is a bar above the set name: P or M __ Monitor gamut (M) Printer gamut (P) Complement sets 1

21 Complement sets 2 U A A B B _

22 Complement sets 3 Formal definition for the complement of a set: A = { x | x  A } –Or U – A, where U is the universal set Further examples (assuming U = Z) –{1, 2, 3} = { …, -2, -1, 0, 4, 5, 6, … } –{New York, Washington} - {3, 4} = {New York, Washington} –{1, 2} -  = {1, 2} The difference of any set S with the empty set will be the set S

23 Properties of complement sets –A = AComplementation law –A U A = U Complement law –A ∩ A =  Complement law Complement sets 4 ¯ ¯ ¯ ¯

24 Quick survey I understand the various set operations I understand the various set operations a) Very well b) With some review, I’ll be good c) Not really d) Not at all

25 A last bit of color…

26 Set identities Set identities are basic laws on how set operations work –Many have already been introduced on previous slides Just like logical equivalences! –Replace U with  –Replace ∩ with  –Replace  with F –Replace U with T Full list on Rosen, page 89

27 Set identities: DeMorgan again These should look very familiar…

28 How to prove a set identity For example: A∩B=B-(B-A) Four methods: –Use the basic set identities (Rosen, p. 89) –Use membership tables –Prove each set is a subset of each other This is like proving that two numbers are equal by showing that each is less than or equal to the other –Use set builder notation and logical equivalences

29 What we are going to prove… A∩B=B-(B-A) AB A∩BB-AB-(B-A)

30 Definition of difference DeMorgan’s law Complementation law Distributive law Complement law Identity law Commutative law Proof by using basic set identities Prove that A∩B=B-(B-A)

31 The top row is all elements that belong to both sets A and B –Thus, these elements are in the union and intersection, but not the difference The second row is all elements that belong to set A but not set B –Thus, these elements are in the union and difference, but not the intersection What is a membership table Membership tables show all the combinations of sets an element can belong to –1 means the element belongs, 0 means it does not Consider the following membership table: The third row is all elements that belong to set B but not set A –Thus, these elements are in the union, but not the intersection or difference The bottom row is all elements that belong to neither set A or set B –Thus, these elements are neither the union, the intersection, nor difference

32 Proof by membership tables The following membership table shows that A∩B=B-(B-A) Because the two indicated columns have the same values, the two expressions are identical This is similar to Boolean logic!

33 Proof by showing each set is a subset of the other 1 Assume that an element is a member of one of the identities –Then show it is a member of the other Repeat for the other identity We are trying to show: –(x  A∩B→ x  B-(B-A))  (x  B-(B-A)→ x  A∩B) –This is the biconditional! Not good for long proofs Basically, it’s an English run-through of the proof

34 Proof by showing each set is a subset of the other 2 Assume that x  B-(B-A) –By definition of difference, we know that x  B and x  B-A Consider x  B-A –If x  B-A, then (by definition of difference) x  B and x  A –Since x  B-A, then only one of the inverses has to be true (DeMorgan’s law): x  B or x  A So we have that x  B and (x  B or x  A) –It cannot be the case where x  B and x  B –Thus, x  B and x  A –This is the definition of intersection Thus, if x  B-(B-A) then x  A∩B

35 Proof by showing each set is a subset of the other 3 Assume that x  A∩B –By definition of intersection, x  A and x  B Thus, we know that x  B-A –B-A includes all the elements in B that are also not in A not include any of the elements of A (by definition of difference) Consider B-(B-A) –We know that x  B-A –We also know that if x  A∩B then x  B (by definition of intersection) –Thus, if x  B and x  B-A, we can restate that (using the definition of difference) as x  B-(B-A) Thus, if x  A∩B then x  B-(B-A)

36 Proof by set builder notation and logical equivalences 1 First, translate both sides of the set identity into set builder notation Then massage one side (or both) to make it identical to the other –Do this using logical equivalences

37 Proof by set builder notation and logical equivalences 2 Original statement Definition of difference Negating “element of” Definition of difference DeMorgan’s Law Distributive Law Negating “element of” Negation Law Identity Law Definition of intersection

38 Proof by set builder notation and logical equivalences 3 Why can’t you prove it the “other” way? –I.e. massage A∩B to make it look like B-(B-A) You can, but it’s a bit annoying –In this case, it’s not simplifying the statement

39 Quick survey I understand (more or less) the four ways of proving a set identity I understand (more or less) the four ways of proving a set identity a) Very well b) With some review, I’ll be good c) Not really d) Not at all

40 Today’s demotivators

41 Computer representation of sets 1 Assume that U is finite (and reasonable!) –Let U be the alphabet Each bit represents whether the element in U is in the set The vowels in the alphabet: abcdefghijklmnopqrstuvwxyz The consonants in the alphabet: abcdefghijklmnopqrstuvwxyz

42 Computer representation of sets 2 Consider the union of these two sets:  Consider the intersection of these two sets: 

43 Rosen, section 1.7 question 14 Let A, B, and C be sets. Show that: a)(AUB)  (AUBUC) b)(A∩B∩C)  (A∩B) c)(A-B)-C  A-C d)(A-C) ∩ (C-B) = 

44 Quick survey I felt I understood the material in this slide set… I felt I understood the material in this slide set… a) Very well b) With some review, I’ll be good c) Not really d) Not at all

45 Quick survey The pace of the lecture for this slide set was… The pace of the lecture for this slide set was… a) Fast b) About right c) A little slow d) Too slow